The Time Value of Money
Future Value - Amount to which an investment will grow after earning interest. Compound Interest - Interest earned on interest. Simple Interest - Interest earned only on the original investment.
Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Future Years 1 2 3 4 5 Interest Earned 6 6 6 6 6 Value 100 106 112 118 124 130 Value at the end of Year 5 = $130
Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year s balance. Today Future Years 1 2 3 4 5 Interest Earned 6.00 6.36 6.74 7.15 7.57 Value 100 106.00112.36119.10126.25133.82 Value at the end of Year 5 = $133.82
FV $100 ( 1 ) r t Example - FV What is the future value of $100 if interest is compounded annually at a rate of 6% for five years? FV $100 (1.06) 5 $133.82
28 30 26 24 22 4 6 8 10 12 14 16 18 20 2 7000 6000 5000 4000 3000 2000 1000 0 Interest Rates 0% 5% 10% 15% Number of Years 0 FV of $100
Peter Minuit bought Manhattan Island for $24 in 1626. Was this a good deal? To answer, determine $24 is worth in the year 2015, compounded at 8%. FV 389 $24 (1.08) $241.10 trillion FYI - The value of Manhattan Island land is well below this figure.
Present Value = PV PV = Future Value after t periods (1+r) t
Example You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years? PV 3000 2 (1.08) $2,572
The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable. PV FV 1r t ( 1 )
Example Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money is 8%, which do you prefer? Total Immediate PV PV 1 2 PV payment 4,000 (1.08) 4,000 (1.08) 1 2 8,000.00 3,703.70 3,429.36 $15,133.06
PVs can be added together to evaluate multiple cash flows. PV C C 1 2... 1 ( 1r ) ( 1r ) 2
Perpetuity A stream of level cash payments that never ends. Annuity Equally spaced level stream of cash flows for a limited period of time.
PV of Perpetuity Formula C = cash payment r = interest rate PV C r
Example - Perpetuity In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%? PV 100, 000 $1, 000, 000. 10
Example - continued If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today? PV 1000000,, $751, 315 1 10 3 (. )
PV of Annuity Formula PV C 1 1 r r( 1r) t C = cash payment r = interest rate t = Number of years cash payment is received
PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years. 1 1 ( 1 ) PVAF r r r t
Example - Annuity You are purchasing a car. You are scheduled to make 3 annual installments of $4,000 per year. Given a rate of interest of 10%, what is the price you are paying for the car (i.e. what is the PV)? PV 4000, 1 1. 10. 10( 1. 10) 3 PV $9, 947. 41
Applications Value of payments Implied interest rate for an annuity Calculation of periodic payments Mortgage payment Annual income from an investment payout Future Value of annual payments FV C PVAF ( 1r) t
Example - Future Value of annual payments You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account? FV 1 1. 10. 10( 1. 10) 4, 000 ( 1. 10) 20 20 FV $229, 100
NPV = PV(future cash flows) Inv 0 Accept investments that have positive NPV Reject investments with negative NPV Mutual exclusive projects
Example Suppose we can invest $50 today and receive $60 in one year. Should we accept the project given a 10% expected return? 60 NPV = -50+ $4.55 1.10
T T NPV = C C1 C2 C 0... 2 (1 IRR) (1 IRR) (1 IRR) 0 IRR IS THE DISCOUNT RATE FOR WHICH NPV=0
FINANCIAL CALCULATOR. TRIAL AND ERROR EXAMPLE: C 0 = - 4,000 C 1 = +2,000 C 3 = +4,000 TRY IRR = 0, NPV = +2,000, IRR > 0 TRY IRR = 50%, NPV = - 889, IRR < 50 TRY IRR = 25%, NPV = +160, IRR >25 TRY IRR = 28%, NPV = 0
ACCEPT PROJECT IF IRR IS GREATER THAN THE OPPORTUNITY COST OF CAPITAL LOOKING AT THE NET PRESENT VALUE PROFILE FOR A CONVENTIONAL PROJECT, WE WILL BE ACCEPTING PROJECTS WITH POSITIVE NPV
CASH OUTFLOWS FOLLOWED BY CASH INFLOWS NPV DECLINES WITH INCREASING DISCOUNT RATES The optimality of NPV rule
NPV Year: 0 1 IRR(%) At 10% ($) A -1,000 +1,500 +50 +364 B +1,000-1,500 +50-364 BOTH PROJECTS HAVE IRR OF 50% NPV PROFILE FOR PROJECT B INCREASES WITH INCREASING DISCOUNT RATES ACCEPT PROJECT B WHEN IRR IS LESS THAN THE OPPORTUNITY COST OF CAPITAL